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Chapter 3: Problem 43

Determine which, if any, of the three given statements are equivalent. You may use information about a conditional statement's converse, inverse, or contrapositive, De Morgan's laws, or truth tables. a. If the grass turns yellow, you did not use fertilizer or water. b. If you use fertilizer and water, the grass will not turn yellow. c. If the grass does not turn yellow, you used fertilizer and water.

Short Answer

Expert verified
Statements a and b are logically equivalent.

Step by step solution

01

Transform into standard form

First step is to transform the statements into the standard logical form (if p then q). After this transformation the sentences are: a. If the grass turns yellow (p), then you did not use fertilizer or water (q). b. If you use fertilizer and water (p), then the grass will not turn yellow (q). c. If the grass does not turn yellow (p), then you used fertilizer and water (q).
02

Formulate contrapositives

In an attempt to find equivalences, formulate the contrapositive of each statement. The contrapositive of (p 鈫� q) is (鈭紂 鈫� 鈭紁). This results in: a. If you did use fertilizer or water, then the grass does not turn yellow. b. If the grass does turn yellow, then you did not use fertilizer and water. c. If you did not use fertilizer and water, then the grass turns yellow.
03

Compare statements

Finally, compare the original statements and their contrapositives to determine if any are logically equivalent. From this comparison, it can be observed that the contrapositive of statement a is similar to statement b, and the contrapositive of statement b is similar to statement a. Therefore, statements a and b are logically equivalent.

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